Traveling plane wave explained

In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed

c , along a fixed direction of propagation \vec n .

Such a field can be written as

F(\vec x, t) = G\left(\vec x ⋅ \vec n - c t\right) where G(u) is a function of a single real parameter u = d - c t . The function G describes the profile of the wave, namely the value of the field at time t = 0 , for each displacement d = \vec x ⋅ \vec n . For each displacement d , the moving plane perpendicular to \vec n at distance d + c t from the origin is called a wavefront. This plane too travels along the direction of propagation \vec n with velocity c ; and the value of the field is then the same, and constant in time, at every one of its points.

The wave

F may be a scalar or vector field; its values are the values of G .

A sinusoidal plane wave is a special case, when

G(u) is a sinusoidal function of u .

Properties

A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector

\vec n ; that is, by considering the wave F(z\vec n,t) = G(z - ct) on a one-dimensional medium, with a single position coordinate z .

For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction

\vec n ; specifically, \nabla F(\vec x,t) = \vec n G'(\vec x ⋅ \vec n - ct) , where G' is the derivative of G . Moreover, a traveling plane wave F of any shape satisfies the partial differential equation\nabla F =- (\vec n)/(c) (\partial F)/(\partial t)

Plane traveling waves are also special solutions of the wave equation in an homogeneous medium.

See also

• Spherical wave
• Standing wave